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Determination of convection terms and quasi-linearities appearing in diffusion equations. (arXiv:1812.08495v1 [math.AP])
来源于:arXiv
We consider the highly nonlinear and ill posed inverse problem of determining
some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion
equation $\partial_tu-\Delta_x u+F(x,t,u,\nabla_xu)=0$ on $\Omega\times(0,T)$,
with $T>0$ and $\Omega$ a bounded open subset of $\mathbb R^n$, $n\geq2$, from
measurements of solutions on the lateral boundary $\partial\Omega\times(0,T)$.
We consider both linear and nonlinear expression of $F(x,t,\nabla_xu,u)$. In
the linear case, the equation is a convection-diffusion equation and our
inverse problem corresponds to the unique recovery, in some suitable sense, of
a time evolving velocity field associated with the moving quantity as well as
the density of the medium in some rough setting described by non-smooth
coefficients on a Lipschitz domain. In the nonlinear case, we prove the
recovery of more general quasilinear expression appearing in a non-linear
parabolic equation. Our result give a positive answer to the unique recovery of
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